GRADED RADICAL W TYPE LIE ALGEBRAS I

(1)

PII. S016117120210812X http://ijmms.hindawi.com

GRADED RADICAL W TYPE LIE ALGEBRAS I

KI-BONG NAM

Received 24 August 2001 and in revised form 28 January 2002

We get a newZ-graded Witt type simple Lie algebra using a generalized polynomial ring which is the radical extension of the polynomial ringF[x]with the exponential functione^x. 2000 Mathematics Subject Classiﬁcation: 17B20.

1. Introduction. LetFbe a field of characteristic zero (not necessarily algebraically closed). Throughout this paper,Z+ and Zdenote the nonnegative integers and the integers, respectively. LetF[x]be the polynomial ring in indeterminatex. LetF(x)= {f (x)/g(x)|f (x), g(x)∈F [x], g(x)≠0}be the field of rational functions in one variable. We define theF-algebraV^√m,espanned by

e^dxf₁^a¹^/b¹···f_m^a^m^/b^mx^t|d, a1, . . . , am, t∈Z, fi≠x, a1, b1

=1, . . . , am, bm

=1, 1≤i≤m ,

(1.1)

whereb1, . . . , bmare ﬁxed nonnegative integers, and(ai, bi)=1,1≤i≤m, means that aiandbiare relatively primes, andf1, . . . , fnare the ﬁxed relatively prime polynomials inF[x]. TheF-subalgebraV^√⁺_{m, e}ofV^√m,eis spanned by

e^dxf₁^a¹^/b¹···f_m^a^m^/b^mx^t|d, a1, . . . , am∈Z, t∈Z+, fi≠x, a1, b1

=1, . . . , am, bm

=1, 1≤i≤m .

(1.2)

LetW^√m,e(∂)be the vector space over Fwith elements{f ∂|f ∈V^√m,e}and the standard basis{e^dxf₁â¹^/b¹···fmâ^m^/b^mx^t∂|e^dxf₁â¹^/b¹···fmâ^m^/b^mx^t∂∈W^√m,e}. Define a Lie bracket onW^√m,e(∂)as follows:

[f ∂, g∂]=f

∂(g)

∂−g

∂(f )

∂, f , g∈V^√m,e. (1.3) It is easy to check that (1.3) deﬁnes a Lie algebraW^√m,e(∂)with the underlying vector spaceW^√m,e(∂)(see also [1,3,5]). Similarly, we deﬁne the Lie subalgebraW^√⁺_m,e(∂)of W^√m,e(∂)using theF-algebraV^√⁺_m,einstead ofV^√m,e.

The Lie algebraW^√m,e(∂)has a naturalZ-gradation as follows:

W^√m,e(∂)=

d∈Z

W^√^d_m,e, (1.4)

whereW^√^d_m,eis the subspace of the Lie algebraW^√m,e(∂)generated by elements of the form{e^dxf₁^a¹^/b¹···fm^a^m^/b^mx^t∂|f1, . . . , fn∈F[x], a1, . . . , am, t∈Z, m∈Z+}. We call the subspaceW^√^d_m,ethed-homogeneous component ofW^√m,e(∂).

(2)

We decompose thed-homogeneous componentW^√^d_m,eas follows:

W^√^d_{m, e}=

s₁,...,s_m∈Z

W(d,s₁,...,sm), (1.5)

whereW(d,s₁,...,s_m)is the subspace ofW^√^d_m,espanned by e^dxf₁^s¹^/b¹···f_m^s^m^/b^mx^q∂|q∈Z

. (1.6)

Note thatW(0,0,...,0)is the Witt algebraW (1)as deﬁned in [3].

The two radical-homogeneous componentsW(d,a1,...,am)andW(d,r1,...,rm)are equivalent ifa1−r1, . . . , am−rm∈Z. This deﬁnes an equivalence relation onW^√^d_m,e. Thus we note that the equivalent class ofW(d,a₁,...,am) depends only ona1, . . . , am. From now onW(d,a₁,...,am)will represent the radical homogeneous equivalent class ofW(d,a₁,...,am)

without ambiguity. It is possible to choose the minimal positive integersa1, . . . , amfor the radical homogeneous equivalent componentW(d,a₁,...,am).

We give the lexicographic order on all the radical homogeneous equivalent compo- nentsW(d,a₁,...,a_m)usingZ×Z^m₊.

The radical equivalent homogeneous componentW^√^d_m,ecan be written as follows:

W^√^d_m,e=

(a₁,...,am)∈Z^m₊

W(d,a₁,...,am). (1.7)

Thus for any elementl∈W^√m,e(∂),lcan be written uniquely as follows:

l=

(d,a₁,...,a_m)∈Z×Z^m+

l(d,a₁,...,a_m). (1.8)

For any such elementl∈W^√m,e(∂),H(l)is defined as the number of different homogeneous components oflas in (1.4), andLd(l)as the number of nonequivalent radicald- homogeneous components oflin (1.8). For each basis elemente^dxf₁â¹^/b¹···fmâ^m^/b^mx^t∂ ofW^√m,e(∂)(orW^√⁺_m,e(∂)), define deg_Lie(e^dxf₁â¹^/b¹···fmâ^m^/b^mx^t∂)=t. Since every ele- mentlofW^√m,e(∂)is the sum of the standard basis element, we may define deg_Lie(l) as the highest power of each basis element ofl. Note that the Lie algebraW^√m,e(∂)is self-centralized, that is, the centralizerCl(W^√m,e(∂))of every elementlinW^√m,e(∂)is one dimensional [1]. We find the solution of

1^1/3=y (1.9)

inZ7. Equation (1.9) implies that

1≡y³mod 7. (1.10)

The solutions of (1.10) are 1, 2, or 4. Thus 1^1/3=1, 2, or 4 mod 7. Thus the radical number inZpis not uniquely determined generally. So we may not consider the Lie algebras in this paper over a ﬁeld of characteristicpdiﬀerently from the Lie algebras in [2,3,4]. It is easy to prove that the Lie algebraW(0,...,0)is simple [3].

(3)

2. Main results. We need several lemmas forTheorem 2.5.

Lemma2.1. For any elementlin the(d, a1, . . . , am)-radical-homogeneous compo- nent ofW^√m(∂), and for any elementl1∈W(0,0,...,0),[l, l1]is an element in the(d, a1, . . . , am)-radical homogeneous equivalent component.

The proof ofLemma 2.1is straightforward.

Lemma2.2. A Lie idealIof W^√m,e(∂)which contains∂isW^√m,e(∂).

Proof. LetIbe the ideal in the lemma. The Lie subalgebra which has the standard basis{xⁱ∂|i∈Z+}is simple. LetI be any ideal ofW^√m,e(∂)which contains∂. Then for anyf ∂∈W^√m,e(∂),

[x∂, f ∂]=x∂(f )∂−f ∂∈I. (2.1)

On the other hand,

[∂, xf ∂]=f ∂+x∂(f )∂∈I. (2.2)

Thus by subtracting (2.2) from (2.1) we get 2f ∂∈I. Therefore, we have proven the lemma, sinceI∩W(0,0,...,0)contains nonzero elements and soI⊃W(0,0,...,0).

Lemma 2.3. A Lie ideal I of W^√m,e(∂) which contains a nonzero element in W(d,a₁,...,am)isW^√m,e(∂), for a fixed(d, a1, . . . , am)∈Z×Z+.

Proof. LetI be a Lie ideal ofW^√m,e(∂) and la nonzero element in the idealI.

Then we take an elementl1=e⁻^dxf₁^−a¹^/b¹···fm⁻^a^m^/b^mx^p∂withpa suﬃciently large positive integer such that[l, l1]≠0. Then[f ∂, [l, l1]]is a nonzero element inW(0,0,...,0)

by taking an elementf₁^t¹···fm^t^m∈F[x], wheret1, . . . , tmare suﬃciently large integers.

ThusI∩W(0,0,...,0)contains nonzero elements, and hence,∂∈I∩W(0,0,...,0)by simplicity ofW(0,0,...,0). Then the lemma follows fromLemma 2.2.

Throughout this paper,abmeans thatais a number suﬃciently larger thanb.

Lemma2.4. LetI be any nonzero Lie ideal ofW^√m,e(∂). For any nonzero element l∈I, there is an elementx^s∂,s0, such that[x^s∂, l]is the sum of elements inW^√m,e(∂) withdeg_Lie([x^s∂, l]) >0.

Proof. It is straightforward by choosing a suﬃciently large positive integers.

Theorem2.5. The Lie algebraW^√m,e(∂)is simple.

Proof. LetIbe a nonzero Lie ideal ofW^√m,e(∂). Letlbe a nonzero element ofI.

ByLemma 2.4, we may assume thatlhas polynomial terms with positive powers for each basis element ofl. We prove this theorem in several steps.

Step1. Iflis in the 0-homogeneous component, then the theorem holds. We prove this step, by induction on the numberL0(l)of nonequivalent radical-homogeneous components of the element lofI. If L0(l)is 1 and l∈W(0,0,...,0), then the theorem holds by Lemmas2.2,2.3, and the fact thatW(0,0,...,0)is simple.

(4)

Assume that l∈ W(0,0,...,0,ar,...,am) with ar ≠ 0. If we take an element f₁^h^r^/k^r···

fn^h^m^/k^mx^h^m+1∂such thathrkr, . . . , hnkr and(hr+kr)/kr∈Z+, . . . , (hm+km)/

km∈Z+, then we havel1=[f₁^h^r^/k^r···fm^h^m^/k^mx^h^m+1∂, l]≠0. This implies thatl1is in W (0,0, . . . ,0). Thus we have proven the theorem byLemma 2.2.

By induction, we may assume that the theorem holds forl∈Isuch thatL0(l)=k, for some ﬁxed nonnegative integerk >1. Assume thatL0(l)=k+1. Iflhas aW(0,0,...,0)

radical-homogeneous equivalent component, we takel2∈W(0,0,...,0)such that[l, l2]can be written as follows:[l, l2]=l3+l4wherel3is a sum of nonzero radical-homogeneous components, andl4=f ∂withf∈F[x]. Thus we have the nonzero element

∂, ···, [∂, l]···

=l2∈I (2.3)

which has no terms in the homogeneous equivalent componentW(0,0,...,0), where we applied Lie brackets untill2has no terms in the radical homogeneous equivalent com- ponentW(0,0,...,0). Thenl2∈Isuch thatH(l2)≤k. Therefore, we have proven the theorem by Lemmas2.2,2.3, and induction. Iflhas no terms in the radical homogeneous equivalent component(0,0, . . . ,0), thenlhas a term in the radical homogeneous equivalent componentW(0,a₁,...,an). Take an elementl3=f₁^c¹^/p¹···fm^c^m^/p^mx^c^m+1∂such that c1, . . . , cm+1are suﬃciently large positive integers such thatc1+a1∈Z···cm+am∈Z, and which is in a radical homogeneous equivalent componentW(0,a₁,...,am). Then[l3, l]

is nonzero and which has a term in the radical homogeneous equivalent component W(0,0,...,0). So in this case we have proven the theorem by induction.

Step2. Assume thatlis in thed-homogeneous component such that 0≠dand L0(l)=1, then the theorem holds. By taking e⁻^dxx^t∂, we have 0≠[e⁻^dxx^t∂, l]∈ W(0,0,...,0) by taking a suﬃciently large positive integer t. Thus we have proven the theorem byStep 1.

Step 3. If l is the sum of (k−1) nonzero homogeneous components and 0- homogeneous component, then the theorem holds. We prove the theorem by induction on the number of distinct homogeneous components by Steps1and2. Assume that we have proven the theorem whenl has(k−1)radical-homogeneous components. Assume thatlhas terms inW(0,0,...,0). ByStep 1, we have an elementl1∈I, such thatl1=l2+f ∂, wherel2has(k−1)homogeneous components andf∈F[x]. Then 0≠∂, [···, [∂, l1]···]∈Ihas(k−1)homogeneous components, where we applied the Lie bracket until it has no terms inW(0,0,...,0). Therefore, we have proven the theorem by induction.

Assume thatlhas a(k)homogeneous equivalent components. We may assumel has the terms which is in 0≠d-homogeneous component. By taking a suﬃciently large positive integerr, we have[e^−dxx^r∂, l]≠0 and it has(k)homogeneous components with a term in the radical-homogeneous componentW(0,0,...,0). Therefore, we have proven the theorem byStep 3.

Corollary2.6. The Lie algebraW^√⁺_m,e(∂)is simple.

Proof. It is straightforward fromTheorem 2.5without usingLemma 2.4.

Corollary2.7. The Lie subalgebraW^√⁰_m,eof W^√m,e(∂)is simple.

(5)

Proof. It is straightforward fromStep 1ofTheorem 2.5.

Proposition 2.8. For any nonzero Lie automorphism θ of W^√⁺_m,e(∂), θ(∂)= ∂ holds.

Proof. It is straightforward from the relationθ([∂, x∂])=θ(∂)and the fact that W^√⁺_m,e(∂)is self-centralized andZ-graded.

Acknowledgments. The author thanks the referee for the valuable suggestions and comments on this paper. The author also thanks Professor Kawamoto for his comments on radical numbers ofZp.

References

[1] N. Kawamoto, K.-B. Nam, and J. Pakianathan,On generalized Witt algebras in one variable, in preparation.

[2] A. I. Kostrikin and I. R. Safarevic,Graded Lie algebras of finite characteristic, Math. USSR- Izv.3(1970), no. 2, 237–240.

[3] K.-B. Nam,GeneralizedW andHtype Lie algebras, Algebra Colloq.6(1999), no. 3, 329–

340.

[4] ,ModularWandHtype Lie algebras, Bulletin of South Eastern Asian Mathematics, Springer-Verlag, 2002, to appear.

[5] A. N. Rudakov,Groups of automorphisms of infinite-dimensional simple Lie algebras, Math.

USSR-Izv.3(1969), 707–722.

Ki-Bong Nam: Department of Mathematics and Computer Science, University of Wisconsin-Whitewater, Whitewater, WI53190, USA

E-mail address:[email protected]

(6)

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at

http://www .hindawi.com/journals/mpe/. Prospective authors should

submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at

http://

mts.hindawi.com/

according to the following timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Edson Denis Leonel,

Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; [email protected]

Alexander Loskutov,

Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

Hindawi Publishing Corporation http://www.hindawi.com